Q99 / 160 · AP EAMCET 2022 (05 Jul Shift 2) · Mathematics

$\lim _{n \rightarrow x} \sqrt{2}\left[\frac{(2+\sqrt{2})^n+(2-\sqrt{2})^n}{(2+\sqrt{2})-(2-\sqrt{2})^n}\right]=$

  • A
    $2-\sqrt{2}$
  • B
    $2-\sqrt{2}$
  • C
    $1$
  • Correct answer
    D
    $\sqrt{2}$

Solution

For $n \rightarrow \infty$ $\begin{aligned} & \because 2-\sqrt{2} < 1 \\ & \therefore \lim _{n \rightarrow \infty}(2-\sqrt{2})^n \rightarrow 0\end{aligned}$ $\begin{aligned} & =\lim _{n \rightarrow \infty} \sqrt{2}\left[\frac{(2+\sqrt{2})^n+0}{(2+\sqrt{2})^n-0}\right] \\ & =\sqrt{2}\end{aligned}$

Hence, the correct answer is option: 'D'.

Relevant Questions